section 1.5. the graph of a function f is the collection of ordered pairs (x, f (x)) where x is in...
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Section 1.5
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The graph of a function f is the collection of ordered pairs (x, f (x)) where x is in the domain of f.
(2, –2) is on the graph of f (x) = (x – 1)2 – 3.
x
y
4
-4(2, –2)
f (2) = (2 – 1)2 – 3 = 12 – 3 = – 2
f (x) = (x – 1)2 – 3
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x
y
4
-4
The domain of the function y = f (x) is the set of values of x for which a corresponding value of y exists.
The range of the function y = f (x) is the set of values of y which correspond to the values of x in the domain.
Domain
Range
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Example 1
Find the domain and range of the function
f (x) = from its graph.3x
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x
y
– 1
1
The domain is [–3,∞).
The range is [0,∞).
Range
Domain
(–3, 0)
3f x x
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x
y
4
-4
Vertical Line Test
A relation is a function if no vertical line intersects its graph in more than one point.
This graph does not pass the vertical line test. It is not a function.
This graph passes the vertical line test. It is a function.
y = x – 1x = | y – 2|
x
y
4
-4
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A function f is:
• increasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) < f (x2),
• decreasing on an interval if, for any x1 and x2 in the interval, x1 < x2 implies f (x1) > f (x2),
• constant on an interval if, for any x1 and x2 in the interval, f (x1) = f (x2).
• The x-value that a graph changes direction is not in the interval.
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The graph of y = f (x):
• decreases on (– ∞, –1),• constant on (–1, 1),
• increases on (1, ∞).
-4 -2 2 4
2
4
6
x
y
(-1, 2) (1, 2)
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A function value f (a) is called a relative minimum of f if there is an interval (x1, x2) that contains a such that x1 < x < x2 implies
f (a) ≤ f (x).
A function value f (a) is called a relative maximum of f if there is an interval (x1, x2) that contains a such that x1 < x < x2 implies
f (a) ≥ f (x).
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x
y
Relative minimum
Relative maximum
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Graphing Utility: Approximate the relative minimum of the function f(x) = 3x2 – 2x – 1.
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1 1Relative Minimum: , 1
3 3
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A function f is even if for each x in the domain of f, f (–x) = f (x).
x
yf (x) = x2
f (–x) = (–x)2 = x2 = f (x)
f (x) = x2 is an even function.
Symmetric with respect to the y-axis.
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A function f is odd if for each x in the domain of f, f (– x) = – f (x).
x
y
f (x) = x3
f (– x) = (– x)3 = –x3 = – f (x)
f (x) = x3 is an odd function.Symmetric with respect to the origin.
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Example 2
Determine whether the function is even, odd, or neither. Then describe the symmetry.
a. f (x) = x6 – 2x2 + 3
b. g(x) = x3 − 5x
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a. f (x) = x6 – 2x2 + 3
f (-x) = (-x)6 – 2(-x)2 + 3
= x6 – 2x2 + 3
f (x) is an even function and has symmetry with the y-axis.
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b. g(x) = x3 − 5x
g(-x) = (-x)3 – 5(-x)
= −x3 + 5x
= −(x3 – 5x)
= −g(x)
g(x) is an odd function and has symmetry with the origin.
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Zeros of a Function
If the graph of a function of x has an x-intercept (a, 0), then a is a zero of the function.
Definition:
The zeros of a function f of x are the x-values for which f (x) = 0.
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Example 3
Find the zeros of the function
f (x) = 2x2 + 13x – 24.
Solve by factoring.
2x2 + 13x – 24 = 0
(2x + 3)(x – 8) = 0
2x + 3 = 0 or x – 8 = 03
2x 8x
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Average Rate of Change
For a linear graph the rate of change is constant between points and is called the slope of the line.
For a nonlinear graph whose slope changes at each point, the average rate of change between any two points (x1, f (x1)) and
(x2, f (x2)) is the slope on the line through the two points.
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The line through the two points is called the secant line, and the slope of this line is denoted as msec.
Average rate of change
of f from x1 to x2
2 1
2 1
f x f x
x x
change in
change in
y
x
secm
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x1 x2
(x2, f (x2))
(x1, f (x1))Secant Line
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Example 4
Find the average rates of change of
f (x) = x2 – 2x if x1 = -2 and x2 = 1.
2 1
2 1
f x f x
x x
1 2
1 2
f f
1 8
3
3
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Example 5
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Write the height h of the rectangle as a function of x.
The rectangle is bound
by two graphs.
f (x) = -x2 + 4x – 1 and
g(x) = 2.
h = top – bottom
h = f (x) – g(x)
2 4 1 2h x x 2 4 3x x
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Example 6
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Write the length L of the rectangle as a function of y.
The rectangle is bound
by two graphs.
x = ½y2 and
x = 0.
L = right – left
210
2L y
21
2y